Along with raising the tension and fear throughout season 1, Lexi's death was vital to the plot. Throughout season 1, Stefan and Damon try to avoid any attention being drawn to them, especially when it comes from law enforcement.
However, around the time of Vicki's death, suspicions are high and Damon, as per usual, enacts a morally ambiguous plan to save himself and Stefan: frame his behavior on Lexi so that the police no longer suspect him. Though it's an unfortunate, Lexi's death does in fact help the Salvatores from being condemned.
Additionally, Lexi's death continues the rising tensions between Stefan and Damon. At this point in the show, the brothers are definitely in a sort of rivalry wherein Stefan is the good vampire and the hero and Damon is the morally grey and dangerous monster.
Lexi's death only emphasizes these points. Plus, Damon murdering Lexi is a betrayal of Stefan as well because of their close friendship. Therefore, Lexi's death keeps these themes moving until they finally explode at the end of the season.
Although Lexi had a very small role in she has managed to become a fan favorite over the years. One big reason for this is that Lexi returned several times after her death in later seasons of thereby deepening her character and her backstory. Namely, Lexi's return showed that, not only was she Stefan's friend, but she also was his closest confidante when it came to turning his humanity on and off.
Lexi consistently helped Stefan keep his humanity, and even helped Damon despite his difficulties. Plus, she saved Stefan from Hell and greets him when he dies in season 8.
Overall, it seems that has gained so much popularity because she is a kind and generous person. While most vampires can be ruthless and cruel, Lexi shows that it isn't too difficult to be a good vampire, and she continues to be this even after death.
Her connection to Stefan is an admirable one, and even though she was a minor part of as a whole, her goodness made her shine through.
